My texture don't show up
to be honest your files are quite a mess :) it seems you didn't exactly understand UV texturing in Blender Render: it happens at first. Also keep in mind that this (older) engine works completely different from Cycles (newest) and their materials are not the same, should you wish to switch to that engine later.

Here I try to explain what I found in your files: you have 2 different materials (with solid grey/brown colors) that you probably wish to use for texturing different part of your object. Indeed you assigned each of those materials to different faces of your object.

The brown material has no texture, but it's been assigned to the table legs geometry:





The grey material has a texture, and it's been assigned to the table top geometry:





As it is, here it how it looks when viewed through different viewport modes:


Texture





Material





Rendered




In the above results, the pink color means that in those parts a texture was expected, but couldn't be found. Of course you didn't upload (or pack) your image with the example files, and if I replace it with an example I created, here's what happens:







Now it seems that Texturing mode can see the right (red) color of my exampel texture, but look at what its UV map sees in the upper left corner:



Nothing!

Here is how you can correct it: I added my texture to the "legs" material, and set it to UV mapping, and you can see that the mapping works.



Then I moved to the "top" material, and corrected its mapping to UV (it was "object"):





And also looks good in render mode:



Ok but what about the Texturing mode, which still can't see the texture for the "legs" part? Here is what I tried: I went to wireframe mode, deselected all, and then selected only "legs" geometry (I used the help from the brown material, pressing the "select" button just below it)



And you can see it selects the right UV vertices, but the image assigned to that UV map is wrong => TBLTXT, so I replaced it selecting my image:



and now it looks correct even in the Textured mode:



That's the end, and I hope it's a bit more clear to you, now. You could also use a single material, assign the UV to it (image + map) and map different part of the image to different part of the geometry, but that's maybe another test you can do!